If anyone knows what is an explanation for this phenomenon, I would be most grateful for your help.
First, Rostov-on-Don, (the nearest large city I could find sunlight data for) has sunnier summers than New York. From NOAA via Wikipedia, Rostov has 286/314/293 hours of sunlight in June/July/August. New York has 257/268/268. This is largely because New York sees two more rainy days per month (and about twice as much rain on those rainy days) during the summer.
Sunlight intensity at a certain latitude is directly related to the angle of incidence of the sun. The declination of the sun is the relative angle from the sun to the horizon given your latitude and the time of year. The sun moves (well the Earth moves...) from the Tropic of Cancer to the Tropic of Capricorn each year, from 23.45 N to 23.45 S. During the northern hemisphere summer solstice, the sun is directly above the Tropic of Cancer. From New York, the angle of incidence of the sun is thus 41.07 N - 23.45 N = 17.62 degrees. During an equinox, when the sun is directly over the equator, the angle of incidence of the sun is 41.07 N - 0 = 41.07 degrees. The cosine of the angle of incidence is the percentage of full equatorial sunlight energy that you would receive at that latitude. Thus, in mid-summer in NY, you get cos(17.62) = 0.953 of maximum possible sunlight; at the equinox you get cos(41.07) = 0.754; at the winter solstice you get cos(64.52) = 0.430.
For any point in the Northern Hemisphere outside of the tropics, sunlight intensity will always be greater at all times of the year at lower latitude. Thus, New York always has greater sunlight intensity than Ukraine...but the difference is not very significant. At summer solstice NY gets 0.953 of the max sunlight, but Ukraine gets cos(23.3) = 0.918 of max sunlight. This is the difference between early July and mid-August in NY; not very significant. Given the close latitudes involved, you would expect to have gotten more net sun in the summer as a child than you do now due to fewer cloudy days. Since this is not what you observe, there must be some other explanation.
Secondly, I grew up near Boston, and my mom made me wear sunscreen in the summer. I used to get sunburns pretty bad. Now I'm 35 and I live in Virginia and I never wear sunscreen, even when I get to the beach. I don't remember the last time that I got a sunburn. Things change, I guess? This is a purely anecdotal observation, but then so is yours. Without any data to measure the changes (maybe I am smarter about being in shade in my older age?) all I can offer is guesses.
I think you should be looking at either your skin or your behavior as the cause of this change, not necessarily at the climate.